While reading the book "Adiabatic Perturbation Theory in Quantum Dynamics" by Teufel, I came up with the following notations:
$$P=\int_{\mathbb R^{3l}}^\oplus dx P_(x)$$ I searched Wikipedia and found out that this notation is about the "direct integral". Now I know the definition of the direct integral, but I want some introductory references that treat the direct integral in a systematic way.
$$L^2(\mathbb R) \otimes H = L^2(\mathbb R, H) = \int_{\mathbb R}^\oplus dx H$$ where $H$ is a Hilbert space. Regarding the above statement, I have the following questions: (1) What is the definition of $L^2(\mathbb R, H)$? I susepct that this space consists of functions $f: \mathbb R \to H$ such that $x\mapsto \langle \psi, f(x)\phi\rangle$ is measurable for all $\psi, \phi \in H$ and $$\int_{\mathbb R} ||f(x)||_H^2 \:dx <\infty.$$ Is it correct?
(2) Why does the above identification valid?
Also, if there are references that deal with these matter, it would be nice to suggest a few.
